A Composition Formula for the Modified Analytic Function Space Fourier–Feynman Transform

Abstract

Composition formula is one of the most important research topics in functional analysis theory. Various relationships can be obtained using the composition formula. Since the generalized Brownian motion process used in this paper has a non-zero mean function, there are many restrictions on obtaining a composition formula for the modified analytic function space Fourier–Feynman transform. This paper contains an idea of how the composition Formula (9) below is established for the modified analytic function space Fourier–Feynman transform on function space. Using this idea, we are able to solve a problem that had never been solved before.

1. Introduction

The function space $C_{a,b}[0,T]$, induced by generalized Brownian motion, was introduced by J. Yeh in [1,2] and studied extensively in [3,4,5,6,7,8], as shown below:

Let $a\left(t\right)$ be an absolutely continuous real-valued function on $[0,T]$ with $a\left(0\right)=0$, $a^{\prime }\left(t\right)\in L_2[0,T]$ and $b\left(t\right)$ as strictly increasing, continuously differentiable real-valued functions, with $b\left(0\right)=0$ and $b^{\prime }\left(t\right)>0$ for each $t\in [0,T]$. Let $(\mathsf{\Omega },\mathcal{B},P)$ be a probability measure space and let $D=[0,T]$ be a closed interval. A real-valued stochastic process Y on $(\mathsf{\Omega },\mathcal{B},P)$ and D is called a generalized Brownian motion process if $Y(0,\omega )$ = 0 almost everywhere; for $0=t_0<t_1<···<t_n\le T$, the n-dimensional random vector $(Y(t_1,\omega ),\dots ,Y(t_n,\omega ))$ is normally distributed with density function

$$\begin{array}{ll}{W}_n(\overrightarrow{t},\overrightarrow{\eta })& ={\left({\left(2\pi \right)}^n{\displaystyle \prod _{j=1}^n}(b\left(t_j\right)-b\left(t_{j-1}\right))\right)}^{-1/2}\\ & \times exp\left\{-{\displaystyle \frac{1}{2}}{\displaystyle \sum _{j=1}^n}{\displaystyle \frac{{(({\eta }_j-a\left(t_j\right))-({\eta }_{j-1}-a\left(t_{j-1}\right)))}^2}{b\left(t_j\right)-b\left(t_{j-1}\right)}}\right\}\end{array}$$

where $\overrightarrow{\eta }=({\eta }_1,\dots ,{\eta }_n)$, ${\eta }_0=0$ and $\overrightarrow{t}=(t_1,\dots ,t_n)$.

As stated in [1] and [2] (pp. 18–20), the generalized Brownian motion process Y induces a probability measure $\mu$ on the measurable space $({\mathbb{R}}^D,{\mathcal{B}}^D)$, where ${\mathbb{R}}^D$ is the space of all real valued functions $x\left(t\right),t\in D,$ and ${\mathcal{B}}^D$ is the smallest $\sigma$-algebra of subsets of ${\mathbb{R}}^D$, with respect to which all the coordinate evaluation maps $e_t\left(x\right)=x\left(t\right)$ defined on ${\mathbb{R}}^D$ are measurable. The triple $({\mathbb{R}}^D,{\mathcal{B}}^D,\mu )$ is a probability measure space. This measure space is called the function space, induced by the generalized Brownian motion process Y, as determined by $a(·)$ and $b(·)$. From the results of Theorem 14.2 in [2], the probability measure $\mu$ induced by Y, taking a separable version, is supported by $C_{a,b}[0,T]$. We say that $(C_{a,b}[0,T],\mathcal{B}\left(C_{a,b}[0,T]\right),\mu )$ is the function space induced by Y, where $\mathcal{B}\left(C_{a,b}[0,T]\right)$ is the Borel $\sigma$-algebra of $C_{a,b}[0,T]$. We then complete this function space to obtain $(C_{a,b}[0,T],\mathcal{W}\left(C_{a,b}[0,T]\right),\mu )$, where $\mathcal{W}\left(C_{a,b}[0,T]\right)$ is the set of all measurable subsets of $C_{a,b}[0,T]$. We note that the coordinate process defined by $e_t\left(x\right)=x\left(t\right)$ on $C_{a,b}[0,T]\times [0,T]$ is also the generalized Brownian motion process determined by $a\left(t\right)$ and $b\left(t\right)$.

The concept of the analytic function space Fourier–Feynman transform on function space was initiated by Chang and Skoug, and studied extensively in [3,4,5,7]. Some related topics with the analytic function space Fourier–Feynman transform were also studied and established some relationships in [8,9]. In particular, the composition formula is one of very important subjects in such fields. However, it is not easy to establish via the original form because the generalized Brownian motion has the non-zero mean function $a\left(t\right)$.

In this paper, we define a more modified analytic function space Fourier–Feynman transform via the Gaussian process, which is called the generalized analytic Fourier–Feynman transform on function space. We then establish a composition formula for the modified analytic function space Fourier–Feynman transform via its original form.

The results in this paper are quite a lot more complicated because the generalized Brownian motion used in this paper is nonstationary in time and is subject to a drift $a\left(t\right)$. The generalized Brownian motion can be used to explain the position of the Ornstein–Uhlenbeck process in an external force field [10].

2. Preliminaries and Motivations

In this section, we state some preliminaries and motivations.

2.1. Preliminaries and Definitions

A subset E of $C_{a,b}[0,T]$ is said to be a scale-invariant measurable provided $\rho E\in \mathcal{W}\left(C_{a,b}[0,T]\right)$ for all $\rho >0$, and a scale-invariant measurable set N is said to be a scale-invariant null set provided $\mu \left(\rho N\right)=0$ for all $\rho >0$. A property that holds except on a scale-invariant null set is said to hold scale-invariant almost everywhere.

Let

$$L_{a,b}^2[0,T]=\left\{v:{\int }_0^T{v}^2\left(s\right)db\left(s\right)<\infty \mathrm{and}{\int }_0^T{v}^2\left(s\right)d\left|a\right|\left(s\right)<\infty \right\}$$

where $\left|a\right|\left(t\right)$ denotes the total variation in the function $a(·)$ on the interval $[0,t]$, and let

$$C_{a,b}^{\prime }[0,T]=\left\{w\in C_{a,b}[0,T]:w\left(t\right)={\int }_0^{t}z\left(s\right)db\left(s\right)\mathrm{for}\mathrm{some}z\in L_{a,b}^2[0,T]\right\}.$$

For $w\in C_{a,b}^{\prime }[0,T]$ with $w\left(t\right)={\int }_0^{t}z\left(s\right)db\left(s\right)$ for $t\in [0,T]$, let $D:C_{a,b}^{\prime }[0,T]\to L_{a,b}^2[0,T]$ be defined by the formula

$$Dw=z\left(t\right)={\displaystyle \frac{w^{\prime }\left(t\right)}{b^{\prime }\left(t\right)}}.$$

Then $C_{a,b}^{\prime }[0,T]$ with inner product

$${(w_1,w_2)}_{C_{a,b}^{\prime }}={\int }_0^{T}D{w}_1\left(t\right)D{w}_2\left(t\right)db\left(t\right)$$

is a separable Hilbert space.

We give a remark for the function $a\left(t\right)$.

Remark 1.

The statements below are equivalent.

(i) The mean function a belongs to $C_{a,b}^{\prime }[0,T]$.

(ii) The following condition holds:

$${\int }_0^T|a^{\prime }{\left(t\right)|}^{2}d\left|a\right|\left(t\right)<\infty .$$

(1)

Because the function $a:[0,T]\to \mathbb{R}$ is an absolutely continuous real-valued function on $[0,T]$ with $a\left(0\right)=0$, $a^{\prime }\left(t\right)\in L^2[0,T]$. One can see that the function $a\left(t\right)=t^{2/3}$, $t\in [0,T]$, is not an element in $L_{a,b}^2[0,T]$ even though its derivative is an element of $L_2[0,T]$. Our conditions on $b:[0,T]\to \mathbb{R}$ imply that $0<M_1<b^{\prime }\left(t\right)<M_2$ for all $t\in [0,T]$ and some positive real numbers $M_1$ and $M_2$. For $w_a\left(t\right)={\displaystyle \frac{a^{\prime }\left(t\right)}{b^{\prime }\left(t\right)}}$, we see that

$$a\left(t\right)={\int }_0^t{\displaystyle \frac{a^{\prime }\left(s\right)}{b^{\prime }\left(s\right)}}db\left(s\right)={\int }_0^t{w}_a\left(s\right)db\left(s\right)$$

and

$${\int }_0^T{w}_a^2\left(t\right)d[b\left(t\right)+|a\left|\left(t\right)\right]\le {\displaystyle \frac{M_2}{M_1}}\parallel a^{\prime }{\parallel }_{L^2[0,T]}+{\displaystyle \frac{1}{M_1^2}}{\int }_0^T|a^{\prime }{\left(t\right)|}^{2}d|a|\left(t\right)<\infty$$

by condition (1).

Let ${\left\{e_n\right\}}_{n=1}^{\infty }$ be a complete orthonormal set in $(C_{a,b}^{\prime }[0,T],\parallel ·{\parallel }_{C_{a,b}^{\prime }})$ such that $D{e}_n$’s are of bounded variation on $[0,T]$. For $w\in C_{a,b}^{\prime }[0,T]$ and $x\in C_{a,b}[0,T]$, we define the Paley–Wiener–Zygmund (PWZ) stochastic integral

$${(w,x)}^{\sim }:=\underset{n\to \infty }{lim}{\int }_0^T\sum _{j=1}^n{(w,e_j)}_{C_{a,b}^{\prime }}D{e}_j\left(t\right)dx\left(t\right)$$

if the limits exists; see [5,7]. Then, the PWZ stochastic integral ${(w,x)}^{\sim }$ is defined for s-a.e. $x\in C_{a,b}[0,T]$. If $w\in C_{a,b}^{\prime }[0,T]$ and $x\in C_{a,b}^{\prime }[0,T]$, then ${(w,x)}^{\sim }$ exists and we have ${(w,x)}^{\sim }={(w,x)}_{C_{a,b}^{\prime }}$. Also, if $Dw=z\in L_{a,b}^2[0,T]$ is of bounded variation on $[0,T]$, then the PWZ stochastic integral ${(w,x)}^{\sim }$ equals the Riemann–Stieltjes integral ${\int }_0^{T}z\left(t\right)dx\left(t\right)$ for $\mu$-a.e. $x\in C_{a,b}[0,T]$. Furthermore, the PWZ stochastic integral has the expected linearity properties. Finally, for each $w\in C_{a,b}^{\prime }[0,T]$, ${(w,x)}^{\sim }$ is a Gaussian random variable with mean ${(w,a)}_{C_{a,b}^{\prime }}$ and variance ${\parallel w\parallel }_{C_{a,b}^{\prime }}^2$. Hence, for all $w_1,w_2\in C_{a,b}^{\prime }[0,T]$, we have

$${\int }_{C_{a,b}[0,T]}{(w_1,x)}^{\sim }{(w_2,x)}^{\sim }d\mu \left(x\right)={(w_1,w_2)}_{C_{a,b}^{\prime }}+{(w_1,a)}_{C_{a,b}^{\prime }}{(w_2,a)}_{C_{a,b}^{\prime }}.$$

Thus, if $\{w_1,\dots ,w_n\}$ is an orthogonal set in $C_{a,b}^{\prime }[0,T]$, then the Gaussian random variables ${(w_j,x)}^{\sim }$’s are independent.

Throughout this paper, let $\mathbb{C}$, ${\mathbb{C}}_{+}$ and ${\tilde{\mathbb{C}}}_{+}$ denote the set of complex numbers, complex numbers with a positive real part, and non-zero complex numbers with a nonnegative real part, respectively. Furthermore, for each $\lambda \in \mathbb{C}$, ${\lambda }^{1/2}$ denotes the principal square root of $\lambda$, i.e., ${\lambda }^{1/2}$ is always chosen to have nonnegative real part, so that ${\lambda }^{-1/2}={\left({\lambda }^{1/2}\right)}^{-1}$ is in ${\mathbb{C}}_{+}$ for all $\lambda \in {\tilde{\mathbb{C}}}_{+}$.

For each $t\in [0,T]$, let ${\chi }_{[0,t]}$ denote the characteristic function of the interval $[0,t]$ and for $k\in C_{a,b}^{\prime }[0,T]$ with $Dk=h$ and with ${\parallel k\parallel }_{C_{a,b}^{\prime }}^2={\int }_0^T{h}^2\left(t\right)db\left(t\right)>0$, let ${\mathcal{Z}}_k(x,t)$ be the PWZ stochastic integral

$${\mathcal{Z}}_k(x,t):={(D^{-1}\left(h{\chi }_{[0,t]}\right),x)}^{\sim }.$$

(2)

In addition, by [2] (Theorem 21.1), ${\mathcal{Z}}_k(·,t)$ is stochastically continuous in t on $[0,T]$. If $h=Dk$ is of bounded variation on $[0,T]$, then ${\mathcal{Z}}_k(x,t)$ is continuous in t for all $x\in C_{a,b}[0,T]$. Furthermore, if $k\left(t\right)\equiv b\left(t\right)$, then ${\mathcal{Z}}_b(x,t)=x\left(t\right)$.

Let $C_{a,b}^{*}[0,T]$ be the set of functions k in $C_{a,b}^{\prime }[0,T]$ such that $Dk$ is continuous except for a finite number of finite jump discontinuities and is of bounded variation on $[0,T]$. For any $w\in C_{a,b}^{\prime }[0,T]$ and $k\in C_{a,b}^{*}[0,T]$, let the operation ⊙ between $C_{a,b}^{\prime }[0,T]$ and $C_{a,b}^{*}[0,T]$ be defined by

$$w\odot k:=D^{-1}\left(DwDk\right),\mathrm{i}.\mathrm{e}.,D(w\odot k)=DwDk,$$

where $DwDk$ denotes the pointwise multiplication of the functions $Dw$ and $Dk$. One can see that $(C_{a,b}^{*}[0,T],\odot )$ is a commutative algebra with the identity $b(·)$, and for $w\in C_{a,b}^{\prime }[0,T]$ and $k\in C_{a,b}^{*}[0,T]$, it follows that

$$\begin{array}{ll}{(w,{\mathcal{Z}}_k(x,·))}^{\sim }& ={\int }_0^{T}Dw\left(t\right)d\left({\int }_0^{t}Dk\left(s\right)dx\left(s\right)\right)\\ & ={\int }_0^{T}Dw\left(t\right)Dk\left(t\right)dx\left(t\right)={(w\odot k,x)}^{\sim }\end{array}$$

(3)

for s-a.e $x\in C_{a,b}[0,T]$.

Definition 1.

Let $h\in C_{a,b}[0,T]$ and $k\in C_{a,b}^{*}[0,T]$ be given. Let $F:C_{a,b}[0,T]⟶\mathbb{C}$ be such that for each $\lambda >0$, the function space integral

$$J\left(\lambda \right)={\int }_{C_{a,b}[0,T]}F({\lambda }^{-{\displaystyle \frac{1}{2}}}{Z}_k(x,·)+c_{\lambda }h(·))d\mu \left(x\right)$$

exists for all $\lambda >0$ where $c_{\lambda }$ is a real number, which depends on λ. If there exists a function $J^{*}\left(\lambda \right)$ analytic in ${\mathbb{C}}_{+}$ such that $J^{*}\left(\lambda \right)=J\left(\lambda \right)$ for all $\lambda >0$, then $J^{*}\left(\lambda \right)$ is defined to be the modified analytic function space integral of F over $C_{a,b}[0,T]$ with parameter λ, and for $\lambda \in {\mathbb{C}}_{+}$, we write

$$J^{*}\left(\lambda \right)={\int }_{C_{a,b}[0,T]}^{a{n}_{\lambda }^{c_{\lambda }},h,k}F\left(x\right)d\mu \left(x\right).$$

Let $q\ne 0$ be a real number and let F be a functional such that ${\int }_{C_{a,b}[0,T]}^{a{n}_{\lambda }^{c_{\lambda }},h,k}F\left(x\right)d\mu \left(x\right)$ exists for all $\lambda \in {\mathbb{C}}_{+}$. If the following limit exists, we call it the modified analytic function space Feynman integral (AFSFI) of F with parameter q and

$${\int }_{C_{a,b}[0,T]}^{an{f}_q^{c_q},h,k}F\left(x\right)d\mu \left(x\right)=\underset{\lambda \to -iq}{lim}{\int }_{C_{a,b}[0,T]}^{a{n}_{\lambda }^{c_{\lambda }},h,k}F\left(x\right)d\mu \left(x\right)$$

where λ approaches $-iq$ through values in ${\mathbb{C}}_{+}$.

Now, we define the modified analytic function space Fourier–Feynman transform (AFSFFT).

Definition 2.

Let $h\in C_{a,b}[0,T]$ and $k\in C_{a,b}^{*}[0,T]$ be given. For $\lambda \in {\mathbb{C}}_{+}$ and $y\in C_{a,b}[0,T]$, let

$$T_{\lambda ,k}^{c_{\lambda },h}\left(F\right)\left(y\right)={\int }_{C_{a,b}[0,T]}^{a{n}_{\lambda }^{c_{\lambda }},h,k}F(x+y)d\mu \left(x\right).$$

For any non-zero real number q, we define the modified AFSFFT $T_{q,k}^{c_q,h}\left(F\right)$ of F by the formula

$$T_{q,k}^{c_q,h}\left(F\right)\left(y\right)=\underset{\lambda \to -iq}{lim}{T}_{\lambda ,k}^{c_{\lambda },h}\left(F\right)\left(y\right)$$

(4)

if it exists.

2.2. Motivation

The concept of AFSFFT on function space was introduced by Chang and Skoug. Unifying constructions and formulas of the AFSFFT on function space have been studied and developed in many works, including [4,6,8,9]. Many studies have previously attempted to obtain the composition formula with respect to the AFSFFT. For a non-zero real number, $q_1$ and $q_2$ with $q_1+q_2\ne 0$

$$T_{q_1}\circ T_{q_2}=T_{q_3}$$

(5)

For an appropriate non-zero real number $q_3$, where $T_q$ is the analytic function space Fourier–Feynman transform defined by formula (roughly definition)

$$T_q\left(F\right)\left(y\right)=\underset{\lambda \to -iq}{lim}{\int }_{C_{a,b}[0,T]}F({\lambda }^{-{\displaystyle \frac{1}{2}}}x+y)d\mu \left(x\right).$$

In a recent paper [5], the authors studied a generalized version of AFSFFT via the Gaussian process. They also tried to establish the composition formula

$$T_{q_1,k_1}\circ T_{q_2,k_2}=T_{q_3,k_3}$$

(6)

where $T_{q,k}$ is the generalized AFSFFT defined by formula (roughly definition)

$$T_{q,k}\left(F\right)\left(y\right)=\underset{\lambda \to -iq}{lim}{\int }_{C_{a,b}[0,T]}F({\lambda }^{-{\displaystyle \frac{1}{2}}}{Z}_k(x,·)+y)d\mu \left(x\right).$$

However, establishing these composition Formulas (5) and (6) are complicated by the non-zero mean function $a\left(t\right)$ of generalized Brownian motion. Various attempts have been made to solve this problem, such as the translation operator and the Rotation process [5]. But it is not possible to express its original form. To address these issues related to function space $C_{a,b}[0,T]$, a modified AFSFFT $T_q^{c_q,h}$ of functionals defined on $C_{a,b}[0,T]$ is presented, along with fundamental formulas related to the inverse transform and the Fubini theorem in [7]. Nevertheless, the composition formula via the Gaussian process has yet to be established.

Therefore, a new method is needed to solve this problem, and we find that it could be solved through a revised definition via the Gaussian process.

3. Main Theorem

Let $k_1$ and $k_2$ be functions in $C_{a,b}^{*}[0,T]$ with $D{k}_j=h_j,j\in \{1,2\}$. Then, there exists a function $\mathbf{s}$ in $C_{a,b}^{*}[0,T]$, such that

$${\left(D\mathbf{s}\left(t\right)\right)}^2={\left(h_1\left(t\right)\right)}^2+{\left(h_2\left(t\right)\right)}^2={\left(D{k}_1\left(t\right)\right)}^2+{\left(D{k}_2\left(t\right)\right)}^2$$

(7)

for $m_{\left|a\right|,b}$-a.e. $t\in [0,T]$. However the function ‘$\mathbf{s}$’ satisfying (7) is not unique. But we will use the symbol $\mathbf{s}(k_1,k_2)$ to denote the function ‘$\mathbf{s}$’ satisfying (7) as above. Also, we let

$$\mathcal{A}(k_1,k_2)\left(t\right)={\int }_0^t(D{k}_1\left(u\right)+D{k}_2\left(u\right)-D\mathbf{s}(k_1,k_2)\left(u\right))da\left(u\right).$$

To simplify expression for results and formulas, we need some notations. For all non-zero real numbers $p_1,\dots ,p_n$ and $h_j\in C_{a,b}[0,T],j=1,2,\dots ,n$, we have

$$\sum _{j=1}^n{p}_j{h}_j\left(t\right)=\gamma h_{\gamma ,\overrightarrow{h}}\equiv \gamma \left(\overrightarrow{p}\right)h_{\gamma \left(\overrightarrow{p}\right),\overrightarrow{h}}$$

for any $\overrightarrow{p}=(p_1,\dots ,p_n)\in {\mathbb{R}}^n$ and $\overrightarrow{h}=(h_1,\dots ,h_n)\in C_{a,b}^n[0,T]$ with $\overrightarrow{p}\ne 0$, where

$$\gamma =\gamma \left(\overrightarrow{p}\right)=\sqrt{p_1^2+\cdots +p_n^2}$$

and

$$h_{\gamma ,\overrightarrow{h}}\left(t\right)=\sum _{j=1}^n{\displaystyle \frac{p_j{h}_j\left(t\right)}{\gamma }}$$

Furthermore, $\gamma$ is a real number and $h_{\gamma ,\overrightarrow{h}}$ is still in $C_{a,b}[0,T]$.

Stating the main theorem, we need a lemma and some notations; see [5].

Lemma 1.

Let F be an μ-integrable functional and let $k_1,k_2\in C_{a.b}^{*}[0,T]$. Then we have the following formula

$$\begin{gathered} {\int }_{C_{a,b}[0,T]}{\int }_{C_{a,b}[0,T]}F(Z_{k_1}(x,·)+Z_{k_2}(y,·))d\mu \left(x\right)d\mu \left(y\right) \\ ={\int }_{C_{a,b}[0,T]}F(Z_{s(k_1,k_2)}(w,·)+\mathcal{A}(k_1,k_2)(·))d\mu \left(w\right). \end{gathered}$$

(8)

The following theorem is the main results in this paper. We shall establish the composition formula for the AFSFFT via its original forms without any other concepts.

Theorem 1.

(Main theorem) Let $k_1,k_2\in C_{a,b}^{*}[0,T]$. Then for all non-zero real numbers q,

$$T_{q,k_1}^{c_q,h_1}\left(T_{q,k_2}^{c_q,h_2}\left(F\right)\right)\left(y\right)=T_{q,\mathbf{s}(k_1,k_2)}^{\gamma \left(\overrightarrow{p}\right),h_{\gamma \left(\overrightarrow{p}\right),\overrightarrow{h}}}\left(F\right)\left(y\right)=T_{q,k_2}^{c_q,h_2}\left(T_{q,k_1}^{c_q,h_1}\left(F\right)\right)\left(y\right)$$

(9)

for s-a.e. $y\in C_{a,b}[0,T]$, where

$$\left\{\begin{array}{c}\overrightarrow{p}=(1,c_q,c_q)\hfill \\ \overrightarrow{h}=(\mathcal{A}(k_1,k_2),h_1,h_2)\hfill \end{array}\right.$$

if they exist.

We shall introduce some notations as below. For a positive real number c and $h\in C_{a,b}[0,T]$, let $F^c(x)=F(c^{-{\displaystyle \frac{1}{2}}}x)$ and ${\left(F_h\right)}^c\left(x\right)=F(c^{-{\displaystyle \frac{1}{2}}}x+h)$. Then

$${\left(F_{y+c(h_1+h_2)}\right)}^{\lambda }\left(x\right)=F({\lambda }^{-{\displaystyle \frac{1}{2}}}x+y+c(h_1+h_2)).$$

(10)

Proof of Theorem 1.

We shall prove Theorem 1. The existence of all modified AFSFFTs need not be demonstrated through assumptions. Therefore, it suffices to show that there holds the commutative property of operators $T_{q,k_1}^{C_q,h_1}$ and $T_{q,k_2}^{C_q,h_2}$ in Formula (9). For $\lambda >0$, using Equations (4) and (10), we have

$$\begin{array}{l}{T}_{\lambda ,k_1}^{c_{\lambda },h_1}\left(T_{\lambda ,k_2}^{c_{\lambda },h_2}\left(F\right)\right)\left(y\right)\\ ={\int }_{C_{a,b}[0,T]}{\int }_{C_{a,b}[0,T]}F({\lambda }^{-{\displaystyle \frac{1}{2}}}{Z}_{k_1}(x_1,·)+{\lambda }^{-{\displaystyle \frac{1}{2}}}{Z}_{k_2}(x_2,·)\\ +y(·)+c_{\lambda }(h_1+h_2))d\mu \left(x_1\right)d\mu \left(x_2\right)\\ ={\int }_{C_{a,b}[0,T]}{\int }_{C_{a,b}[0,T]}{\left(F_{y+c_{\lambda }(h_1+h_2)}\right)}^{\lambda }(Z_{k_1}(x_1,·)+Z_{k_2}(x_2,·))d\mu \left(x_1\right)d\mu \left(x_2\right).\end{array}$$

Applying Equation (8) to the functional ${\left(F_{y+c(h_1+h_2)}\right)}^{\lambda }$ instead of F, we have

$$\begin{array}{l}{T}_{\lambda ,k_1}^{c_{\lambda },h_1}\left(T_{\lambda ,k_2}^{c_{\lambda },h_2}\left(F\right)\right)\left(y\right)\\ ={\int }_{C_{a,b}[0,T]}F({\lambda }^{-{\displaystyle \frac{1}{2}}}{Z}_{k_3}(w,·)+y(·)+\mathcal{A}(k_1,k_2)(·)+c_{\lambda }(h_1+h_2))d\mu \left(w\right)\\ =T_{\lambda ,k_3}^{c_{\lambda }^{\prime },h_3}\left(F\right)\left(y\right)\end{array}$$

where

$$k_3=\mathbf{s}(k_1,k_2)$$

and

$$c_{\lambda }^{\prime }{h}_3\left(t\right)=\mathcal{A}(k_1,k_2)\left(t\right)+c_{\lambda }(h_1\left(t\right)+h_2\left(t\right)).$$

It can be analytical throughout in ${\mathbb{C}}^{+}$, and so it can let $\lambda \to -iq$. Then, the first equality in Equation (9) is established. We next show that the second equality holds. Using the symmetric property of function space integrals

$${\int }_{C_{a,b}[0,T]}{\int }_{C_{a,b}[0,T]}F(px+qy)d\mu \left(x\right)d\mu \left(y\right)={\int }_{C_{a,b}[0,T]}{\int }_{C_{a,b}[0,T]}F(px+qy)d\mu \left(y\right)d\mu \left(x\right)$$

for all non-zero complex numbers p and q. In a similar method to that used in the first proof above, the second equality is obtained and so we have the desired results. □

Corollary 1.

By using the mathematical induction for n, we have the following assertion

$$\begin{array}{ll}{T}_{q,k_1}^{c_q,h_1}(\cdots \left(T_{q,k_n}^{c_q,h_n}\left(F\right)\right)\cdots )\left(y\right)& =T_{q,\mathbf{s}(k_1,\dots ,k_n)}^{\gamma \left(\overrightarrow{p}\right),h_{\gamma \left(\overrightarrow{p}\right),\overrightarrow{h}}}\left(F\right)\left(y\right)\\ & =T_{q,k_{\pi \left(1\right)}}^{c_q,h_{\pi \left(1\right)}}(\cdots \left(T_{q,k_{\pi \left(n\right)}}^{c_q,h_{\pi \left(n\right)}}\left(F\right)\right)\cdots )\left(y\right)\end{array}$$

for s-a.e. $y\in C_{a,b}[0,T]$, where π is an any permutation of $\{1,2,\dots ,n\}$ and

$$\left\{\begin{array}{c}\overrightarrow{p}=(1,c_q,c_q,\dots ,c_q)\hfill \\ \overrightarrow{h}=(\mathcal{A}(k_1,\dots ,k_n),h_1,\dots ,h_n)\hfill \end{array}\right..$$

Remark 2.

The composition formula for the various transform is a very important formula to establish basic fundamental formulas. As mentioned in Section 2.2, we can obtain various fundamental formulas by using Equation (9).

We next give two examples to illustrate usefulness of the main theorem. From these examples, one can show that $T_{q,k_1}^{c_q,h_1}\left(T_{q,k_2}^{c_q,h_2}\left(F\right)\right)$ is not easy to obtain this. However, using Equation (9), we can evaluate $T_{q,k_1}^{c_q,h_1}\left(T_{q,k_2}^{c_q,h_2}\left(F\right)\right)$ very easily via the single AFSFFT.

Example 1.

For each $w\in C_{a,b}^{\prime }[0,T]$, let

$$F_1\left(x\right)={(w,x)}^{\sim }.$$

Let $k_1\left(t\right)=sint$ and $k_2\left(t\right)=cost$ on $[0,T]$. Then, $D{k}_1\left(t\right)=cost$ and $D{k}_2\left(t\right)=-sint$ and hence $D{\mathbf{s}}^2(k_1,k_2)\left(t\right)={cos}^{2}t+{sin}^{2}t=1$. Thus, $\mathbf{s}(k_1,k_2)\left(t\right)=b\left(t\right)$ and so $Z_{\mathbf{s}(k_1,k_2)}(x,t)=x\left(t\right)$. It is not easy to calculate $T_{q,k_1}^{c_q,h_1}\left(T_{q,k_2}^{c_q,h_2}\left(F_1\right)\right)\left(y\right)$ of $F_1$. However, using Equation (9), we can calculate $T_{q,k_1}^{c_q,h_1}\left(T_{q,k_2}^{c_q,h_2}\left(F_1\right)\right)\left(y\right)$ very easy as follows:

$$\begin{array}{ll}{T}_{q,k_1}^{c_q,h_1}\left(T_{q,k_2}^{c_q,h_2}\left(F_1\right)\right)\left(y\right)& =T_{q,\mathbf{s}(k_1,k_2)}^{\gamma \left(\overrightarrow{p}\right),h_{\gamma \left(\overrightarrow{p}\right),\overrightarrow{h}}}\left(F_1\right)\left(y\right)\\ & ={(w,y)}^{\sim }+{(-iq)}^{-{\displaystyle \frac{1}{2}}}{(w,a)}_{C_{a,b}^{\prime }}+\gamma \left(\overrightarrow{p}\right){(w,h_{\gamma \left(\overrightarrow{p}\right),\overrightarrow{h}})}_{C_{a,b}^{\prime }}.\end{array}$$

Furthermore, if $a\left(t\right)={\displaystyle \frac{1}{2}}{t}^2$ on $[0,T]$, then

$$\mathcal{A}(k_1,k_2)\left(t\right)=sint+cost-t$$

and so

$$\gamma \left(\overrightarrow{p}\right){(w,h_{\gamma \left(\overrightarrow{p}\right),\overrightarrow{h}})}_{C_{a,b}^{\prime }}={(w,sint+cost-t)}_{C_{a,b}^{\prime }}+c_q{(w,sint+cost)}_{C_{a,b}^{\prime }}.$$

Hence, we conclude that

$$\begin{array}{l}{T}_{q,k_1}^{c_q,h_1}\left(T_{q,k_2}^{c_q,h_2}\left(F_1\right)\right)\left(y\right)\\ ={(w,y)}^{\sim }+{(-iq)}^{-{\displaystyle \frac{1}{2}}}{(w,a)}_{C_{a,b}^{\prime }}+{(w,sint+cost-t)}_{C_{a,b}^{\prime }}+c_q{(w,sint+cost)}_{C_{a,b}^{\prime }}.\end{array}$$

Example 2.

For each $w\in C_{a,b}^{\prime }[0,T]$, let

$$F_2\left(x\right)=exp\left\{i{(w,x)}^{\sim }\right\}.$$

Let $b\left(t\right)=t$ on $[0,T]$. Let $k_1\left(t\right)=1$ and $k_2\left(t\right)=ln|sect+tant|$ on $[0,T]$. Then $D{k}_1\left(t\right)=0$ and $D{k}_2\left(t\right)=sect$ and hence $D^2\mathbf{s}(k_1,k_2)\left(t\right)={sec}^{2}t$. Thus, we take $\mathbf{s}(k_1,k_2)\left(t\right)=tant$. Using Equation (9), we can calculate $T_{q,k_1}^{c_q,h_1}\left(T_{q,k_2}^{c_q,h_2}\left(F_2\right)\right)\left(y\right)$ via the single transform as follows:

$$\begin{array}{l}{T}_{q,k_1}^{c_q,h_1}\left(T_{q,k_2}^{c_q,h_2}\left(F_2\right)\right)\left(y\right)=T_{q,\mathbf{s}(k_1,k_2)}^{\gamma \left(\overrightarrow{p}\right),h_{\gamma \left(\overrightarrow{p}\right),\overrightarrow{h}}}\left(F_2\right)\left(y\right)\\ =exp\{{(w,y)}^{\sim }-{\displaystyle \frac{i}{2q}}{\parallel w\odot \mathbf{s}(k_1,k_2)\parallel }_{C_{a,b}^{\prime }}^2+i{\left({\displaystyle \frac{i}{q}}\right)}^{{\displaystyle \frac{1}{2}}}{(w\odot \mathbf{s}(k_1,k_2),a)}_{C_{a,b}^{\prime }}\\ +\gamma \left(\overrightarrow{p}\right){(w,h_{\gamma \left(\overrightarrow{p}\right),\overrightarrow{h}})}_{C_{a,b}^{\prime }}\}.\end{array}$$

Furthermore, if $w\left(t\right)={\displaystyle \frac{2t+sin2t}{4}}$ on $[0,T]$, then $\parallel w\odot \mathbf{s}(k_1,k_2){\parallel }_{C_{a,b}^{\prime }}^2=T^2$ and ${(w\odot \mathbf{s}(k_1,k_2),a)}_{C_{a,b}^{\prime }}=a\left(T\right)$. Hence, we have

$$\begin{array}{l}{T}_{q,k_1}^{c_q,h_1}\left(T_{q,k_2}^{c_q,h_2}\left(F_2\right)\right)\left(y\right)\\ =exp\left\{{(w,y)}^{\sim }-{\displaystyle \frac{i}{2q}}{T}^2+i{\left({\displaystyle \frac{i}{q}}\right)}^{{\displaystyle \frac{1}{2}}}a\left(T\right)+\gamma \left(\overrightarrow{p}\right){(w,h_{\gamma \left(\overrightarrow{p}\right),\overrightarrow{h}})}_{C_{a,b}^{\prime }}\right\}.\end{array}$$

4. Conclusions: Open Question

In Theorem 1, we established the composition Formula (9) for the modified AFSFFT. For a fixed non-zero real number q, the composition of two modified AFSFFTs can be expressed by one modified AFSFFT. Using the similar method in Theorem 1, for fixed function $k\in C_{a,b}^{*}[0,T]$, we can obtain that

$$\begin{array}{ll}{T}_{q_1,k}^{c_{q_1},h_1}\left(T_{q_2,k}^{c_{q_2},h_2}\left(F\right)\right)\left(y\right)& =T_{q_3,k}^{c_{q_3},h_3}\left(F\right)\left(y\right)\\ & =T_{q,k}^{c_{q_2},h_2}\left(T_{q_1,k}^{c_{q_1},h_1}\left(F\right)\right)\left(y\right)\end{array}$$

for s-a.e. $y\in C_{a,b}[0,T]$, where $q_3={\displaystyle \frac{q_1{q}_2}{q_1+q_2}}$ and for some $h_3$. From Examples 1 and 2, we can see that our composition formulas make computation much easier without the need for repetitive calculations. However, the composition formula could not be obtained for non-fixed q and k. There are difficulties in expressing again as the modified AFSFFT in the process of proof, namely

$$T_{q_1,k_1}^{c_{q_1},h_1}\left(T_{q_2,k_2}^{c_{q_2},h_2}\left(F\right)\right)\left(y\right)=T_{q_3,k_3}^{c_{q_3},h_3}\left(F\right)\left(y\right)=T_{q,k_2}^{c_{q_2},h_2}\left(T_{q_1,k_1}^{c_{q_1},h_1}\left(F\right)\right)\left(y\right)$$

(11)

for s-a.e. $y\in C_{a,b}[0,T]$, and some $q_3$ and $k_3$. We are currently researching to solve this difficulty of Equation (11), and we expect to obtain ideas from other research papers.